#9945542

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(18,22)(20,27)(23,29)(25,28)(33,34,38,43)(35,40,46,39)(36,41,47,45)(37,44,48,42);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(18,22)(23,29)(24,30)(31,32)(33,35,38,46)(34,39,43,40)(36,44,47,42)(37,45,48,41);(17,18,21,22)(19,23,26,29)(20,24,27,30)(25,31,28,32)(33,35,38,46)(34,39,43,40)(36,44,47,42)(37,45,48,41);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31)(33,36)(34,41)(35,44)(37,39)(38,47)(40,48)(42,46)(43,45);(1,6,7,3)(2,8,4,10)(5,11,14,13)(9,16,12,15)(17,20)(18,24)(19,25)(21,27)(22,30)(23,31)(26,28)(29,32)(33,37,38,48)(34,42,43,44)(35,45,46,41)(36,40,47,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852738);